New methods of DNA sequencing have produced a great abundance of DNA sequences from humans and many other species, including species now extinct. The proposed research will be intended to help analyze and interpret DNA sequence data both for understanding basic genetic processes such as natural selection and recombination and for understanding the history of humans and the genetic basis of inherited diseases. The proposed research is in four areas: (i) the analysis of DNA sequence data from Neanderthal bones for the purpose of understanding the relationship between Neanderthals and modern humans and for understanding the recent history of modern humans since their separation from Neanderthals, (ii) the prediction of patterns in gene frequencies in populations that have undergone a recent range expansion, with particular emphasis on past range expansions of modern humans populations in Europe, Asia, and the Americas, (iii) the analysis of DNA sequences of bacterial species obtained in samples from extreme environments, for the purpose of quantifying the extent of genetic recombination and differentiation between different strains, (iv) modeling the genetic basis of complex inherited diseases such as schizophrenia, many cancers, and type 1 and 2 diabetes, for the purpose of quantifying the interactions among genes that create higher risk and for devising new methods for identifying genes associated with higher risk. Although the goals of the four areas are somewhat different, similar mathematical and statistical methods will be used in each. Models of population genetic forces, including natural selection, recombination, genetic drift and migration will be developed and tested against available data. Computer simulations of those models will be used to generate hypothetical data sets for testing proposed statistical methods.